Analytical solutions to the compressible Euler equations with time-dependent damping and free boundaries

Abstract

In this paper, we study a class of analytical solutions to the compressible Euler equations with time-dependent damping [Formula: see text], which describe compressible fluids moving into outer vacuum. Under the continuous density condition across the free boundaries separating the fluid from vacuum, we construct a class of spherically symmetric and self-similar analytical solutions in [Formula: see text]. The global-in-time existence of such solutions is proved for μ > 0 and λ > 1. Moreover, the free boundary tends to + ∞ at an algebraic rate not more than C(1 + t)2 as t → + ∞.

Asymptotic Analysis of Systems with Damped Oscillatory Perturbations

“…contain a finite number of harmonics e ik 2 S . Moreover, it assumed that the following condition holds: The evolution of r(t) ≡ x 2 (t) + y 2 (t) for solutions to system (7) with…”

“…where γ(S) = γ 0 + γ 1 sin S, S(t) = √ 2t, λ, γ 0 , γ 1 ∈ R, and p ∈ Z + . It can easily be checked that system (7) corresponds in the polar coordinates x = r cos ϕ, y = −r sin ϕ to (1) with f (r, ϕ, S, t) ≡ t −1 λr sin 2 ϕ + t − p 2 γ(S) sin 2 ϕ, g(r, ϕ, S, t) ≡ t −1 λr sin ϕ cos ϕ + t − p 2 γ(S) sin ϕ cos ϕ,…”

“…Note also that system (7) from Section 1 corresponds to system (55) with α 0 = α 1 = β 1 = γ 0 = s 2 = 0, h = 2 and β 0 = λ. 1.…”

“…The paper is devoted to studying the effect of perturbations decaying in time on the class of autonomous systems. Such perturbed asymptotically autonomous systems arise in a wide range of problems, for example, in the study of epidemiological models [2], Painlevé equations [1], phase-locking phenomena [3], celestial motions [4], and in other problems associated with nonlinear non-autonomous systems [5][6][7]. Note that global properties of solutions of asymptotically autonomous systems have been studied in many papers [8][9][10].…”

Analytical solutions to the isothermal Euler equations with time-dependent damping

“…Later, Sugiyama et al [11,12], Dong et al [13] and Chen et al [14] improved Pan's work. In [15], the authors constructed some radial symmetric solutions with 0  = r or () = u c t r to the 1D compressible Euler equations with time-depending damping. In this paper, we are interested in the construction of non-radially symmetric solutions to compressible isothermal Euler equations with time-depending damping.…”